Let $X_1, X_2, \cdots$ be a sequence of independent random variables with $S_n = \sum^n_{i = 1} X_i$. Fix $\alpha > 0$. Let $\Phi(\cdot)$ be a continuous, strictly increasing function on $\lbrack 0, \infty)$ such that $\Phi(0) = 0$ and $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $x > 0$ and all $c \geq 2$. Suppose $a$ is a real number and $J$ is a finite nonempty subset of the positive integers. In this paper we are interested in approximating $E \max_{j \in J} \Phi(|a + S_j|)$. We construct a number $b_J(a)$ from the one-dimensional distributions of the $X$'s such that the ratio $E \max_{j \in J} \Phi(|a + S_j|)/\Phi(b_J(a))$ is bounded above and below by positive constants which depend only on $\alpha$. Bounds for these constants are given.
Publié le : 1981-06-14
Classification:
Sums of independent random variables,
expectations,
truncated mean,
truncated expectation,
truncated second moment,
tail $\Phi$-moment,
$K$-function,
approximation of expectations,
approximation of integrals,
60G50,
60E15,
60J15
@article{1176994415,
author = {Klass, Michael J.},
title = {A Method of Approximating Expectations of Functions of Sums of Independent Random Variables},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 413-428},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994415}
}
Klass, Michael J. A Method of Approximating Expectations of Functions of Sums of Independent Random Variables. Ann. Probab., Tome 9 (1981) no. 6, pp. 413-428. http://gdmltest.u-ga.fr/item/1176994415/